{
  "id": "1806.05257",
  "version": "v1",
  "published": "2018-06-13T20:28:32.000Z",
  "updated": "2018-06-13T20:28:32.000Z",
  "title": "Distance difference representations of Riemannian manifolds",
  "authors": [
    "Sergei Ivanov"
  ],
  "comment": "25 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "Let $M$ be a complete Riemannian manifold and $F\\subset M$ a set with a nonempty interior. For every $x\\in M$, let $D_x$ denote the function on $F\\times F$ defined by $D_x(y,z)=d(x,y)-d(x,z)$ where $d$ is the geodesic distance in $M$. The map $x\\mapsto D_x$ from $M$ to the space of continuous functions on $F\\times F$, denoted by $\\mathcal D_F$, is called a distance difference representation of $M$. This representation, introduced recently by M. Lassas and T. Saksala, is motivated by geophysical imaging among other things. We prove that the distance difference representation $\\mathcal D_F$ is a locally bi-Lipschitz homeomorphism onto its image $\\mathcal D_F(M)$ and that for every open set $U\\subset M$ the set $\\mathcal D_F(U)$ uniquely determines the Riemannian metric on $U$. Furthermore the determination of $M$ from $\\mathcal D_F(M)$ is stable if $M$ has a priori bounds on its diameter, curvature, and injectivity radius. This extends and strengthens earlier results by M. Lassas and T. Saksala.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-13T20:28:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C20"
    ],
    "keywords": [
      "distance difference representation",
      "strengthens earlier results",
      "complete riemannian manifold",
      "injectivity radius",
      "priori bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}